Poisson-Nijenhuis groupoids

TL;DR

This paper introduces multiplicative Poisson-Nijenhuis structures on Lie groupoids, extending previous concepts, and establishes a correspondence with P-N Lie bialgebroid structures under certain conditions.

Contribution

It defines multiplicative Poisson-Nijenhuis structures on Lie groupoids and links them to P-N Lie bialgebroids, broadening the understanding of compatible geometric structures.

Findings

Establishes a one-to-one correspondence between multiplicative Poisson-Nijenhuis structures and P-N Lie bialgebroids.

Introduces the concept of P-N Lie bialgebroids as a hierarchy of compatible structures.

Extends symplectic-Nijenhuis groupoids to Poisson-Nijenhuis groupoids.

Abstract

We define multiplicative Poisson-Nijenhuis structures on a Lie groupoid which extends the notion of symplectic-Nijenhuis groupoid introduced by Sti\'e23non and Xu. We also introduce a special class of Lie bialgebroid structure on a Lie algebroid $A$, called P-N Lie bialgebroid, which defines a hierarchy of compatible Lie bialgebroid structures on $A$. We show that under some topological assumption on the groupoid, there is a one-to-one correspondence between multiplicative Poisson-Nijenhuis structures on a Lie groupoid and P-N Lie bialgebroid structures on the corresponding Lie algebroid.